Alright Tetrahedron

Geometry Level 5

In a tetrahedron \(ABCD,\) the lengths of \(AB,\) \(AC,\) and \(BD\) are \(6,\) \(10,\) and \(14\) respectively. The distance between the midpoints \(M\) of \(AB\) and \(N\) of \(CD\) is \(4.\) The line \(AB\) is perpendicular to \(AC,\) \(BD,\) and \(MN.\) The volume of \(ABCD\) can be written as \(a\sqrt{b},\) where \(a\) and \(b\) are positive integers, and \(b\) is not divisible by the square of a prime number. What is the value of \(a+b\)?

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